An accurate and time-parallel rational exponential integrator for hyperbolic and oscillatory PDEs

Abstract

Rational exponential integrators (REXI) are a class of numerical methods that are well suited for the time integration of linear partial differential equations with imaginary eigenvalues. Since these methods can be parallelized in time (in addition to the spatial parallelization that is commonly performed) they are well suited to exploit modern high performance computing systems. In this paper, we propose a novel REXI scheme that drastically improves accuracy and efficiency. The chosen approach will also allow us to easily determine how many terms are required in the approximation in order to obtain accurate results. We provide comparative numerical simulations for a shallow water equation that highlight the efficiency of our approach and demonstrate that REXI schemes can be efficiently implemented on graphic processing units.

An exponential integrator/WENO discretization for sonic-boom simulation on modern computer hardware

“…Transverse convection. The last equation (7) is the transverse convection. The solution is obtained by a Lax-Wendroff method, which is conservative and of second order both in σ and ρ.…”

“…Therefore, we set A = 3.4 • 10 −4 . The algorithm proceeds sequentially to solve (3)- (7). In the following we discuss the computational cost of the single steps.…”

“…These values are chosen so that the CFL conditions imposed by the splitting approach and the exponential integrator approach are fulfilled. In the splitting approach the CFL conditions imposed by the Godunov and the Lax-Wendroff scheme, used to solve (4), (7), respectively, read…”